#6554724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6,10,9)(7,12,8,11)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,32)(26,34)(27,36)(29,35)(31,38)(33,37)(39,41)(40,43)(42,44);(5,7,10,8)(6,11,9,12)(13,15,17,27,19,29,18,28)(14,20,23,35,24,36,22,25)(16,26,32,41,33,42,31,40)(21,34,30,43,38,44,37,39);(2,4)(5,6,10,9)(7,12,8,11)(13,16,17,32,19,33,18,31)(14,21,23,30,24,38,22,37)(15,26,27,41,29,42,28,40)(20,34,35,43,36,44,25,39);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,40,42,41)(34,43,44,39);(5,9,10,6)(7,12,8,11)(13,17,19,18)(14,22,24,23)(15,28,29,27)(16,32,33,31)(20,35,36,25)(21,37,38,30)(26,40,42,41)(34,43,44,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461920);